Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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Is the duality product of dual spaces unique?

Let $\Omega \subset \mathbb{R}^n$. Consider, for example, the Sobolev space $H^1_0(\Omega)$. It is known that the dual is $H^{-1}(\Omega)$ and is Banach with respect to the operator norm $$||f||_{-1} =...
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Defining the Fourier projection operator on $H^{-1}$ so that it moves freely on the dual pairing?

Let us consider the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and for any $f \in L^2(\mathbb{T},\mathbb{C})$, define $P_N : L^2(\mathbb{T},\mathbb{C}) \to L^2(\mathbb{T},\mathbb{C})$ as \begin{...
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How can a non-reflexive space have a reflexive subspace?

I know that this can happen (take any one-dimensional subspace, for instance), but I had the following thought while reading Kadison and Ringrose (Fundamentals of the Theory of Operator Algebras, Vol ...
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On the quadratic coalgebras

It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, ...
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Prove that dual of a unital Banach algebra is nonempty using resolvent and Liouville's theorem

Let $\mathcal{A}$ be a unital Banach algebra over complex numbers $\mathbb{C}$. For every $a \in \mathcal{A}$, let $\sigma(a)$ be the spectrum of $a$. Define the resolvent of $a$ to be $ R(a,z) = (...
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The correct way to map a column vector into a row vector. [closed]

I'm a physics student studying linear algebra and the matrix representation of usual and dual vectors. I have several questions related to this topic, and I'm not sure if they make sense. Firstly, the ...
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Image and kernel of transposed maps

Let $U,V,W$ be vector spaces over a field $K$ and $\varphi: U\to V,$ $\psi: V\to W$ linear maps. Given that $\text{ker}(\psi)=\text{im}(\varphi)$ show that $\ker(\varphi^{tr})=\operatorname{im}(\psi^{...
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On the definition of pullback in linear algebra

Here is the definition of pullback maps in linear algebra from Mclnerney's First Steps in Differential Geometry: Let $\Psi: V\to W$ be a linear transformation and let $\Psi^*:W^*\to V^*$ be given by $(...
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Dual space of a topological vector space that doesn't separate points?

While I am studying FUNCTIONAL ANALYSIS by Walter Rudin, I found the following corollary. Now, I wonder how the dual space(the set of all continuous linear functionals on $X$) of some pathetic ...
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Definitions of non degenerate bilinear forms

Referring to this Wikipedia page, The definition of a non degenerate bilinear form is given as a bilinear form $f(x,y)$ such that $v \to (x \to f(x,v))$ is an isomorphism. We are also told that the ...
Siddharth Bhat's user avatar
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Identifying dual space of a hyperplane in Euclidean spaces

I am considering a hyperplane in $\mathbb R^n$ given by \begin{equation} V=\left\{ (v_1,\ldots,v_n)\in\mathbb R^n: \sum_{i=1}^n v_i =0 \right\}. \end{equation} Question: Is there an explicit ...
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How to show duality of $l_1$ and $l_\infty$ balls using linear programming duality.

Let $\| x \|_1 = \sum_{i=1}^n |x_i|$ and $\| x \|_\infty = \max_{i=1}^n |x_i|$ be the $l_1$ and $l_\infty$ norms on $\mathbb{R}^n$. For any given norm on $\mathbb{R}^n$, the associated dual norm (in ...
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What is the relationship between convex conjugate and polar set?

Given a duality $\left<X^*,X\right>$ over field $\mathbb R$, and any set $A \subseteq X$, the polar set of $A$ is defined as \begin{align}A^\circ = \{x^* \in X^* | \left<x^*,x\right> \leq ...
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Why is the transpose so useful?

I am learning linear algebra using the textbook Linear Algebra Done Right, trying to understand the subject through a logical, pure math perspective. I'm, simultaneously, learning applied linear ...
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How do we know $T'(\psi_j)=\psi_j\circ T$, where $T:V\to W$, $T': W'\to W'$ is dual map of $T$, and $\psi_j\in W'$?

Sometimes I have trouble with the notation in the book "Linear Algebra Done Right" by Axler. Currently, my issue is with the following theorem (Theorem 3.114 in the 3rd edition book) Let $...
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What does the transpose of a linear transformation represent? [duplicate]

I'm struggling to understand what the transpose of a linear transformation represents. My textbook's motivation for this wasn't very helpful. All it did was ask, "Is there a linear transformation ...
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Let $M$ be a smooth manifold. Does there exist a canonical isomorphism between $\Gamma(TM)^*$ and $\Gamma(T^*M)$?

Let $M$ be a smooth manifold and for each point $p \in M$, let $T_pM$ denote the tangent space at $p \in M$. We define the set $TM = \bigsqcup_{p \in M}T_pM$ and equip it with the initial topology and ...
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Dimension of range($\Gamma$) where $\Gamma$ is the linear map from $V'\to\mathbb{F}$ defined $\Gamma(\varphi)=(\varphi(v_1),...,\varphi(v_m))$?

Suppose $V$ is finite-dimensional and $v_1,...,v_m\in V$. Define a linear map $\Gamma: V'\to \mathbb{F}^m$ by $$\Gamma(\varphi)=(\varphi(v_1),...,\varphi(v_m))\tag{1}$$ where $V'$ is the dual map of $...
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Is the dual of a unitary operator on a Hilbert space always a continous linear operator?

I have encountered this in a course (not functional analysis), but I am wondering whether this is a general fact. Thank you very much in advance for any input!
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dual norm optimization problem, how to dervie the second formula from the first one?

There is a post that someone answered it already, but I think his/her answer is how to validate the formula, not the way to derive the formula. A dual norm optimization problem This formula comes from ...
containletters's user avatar
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Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta

I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel $$ k \colon X \times X \to \{ 0, 1 \}, \qquad (x, y) \mapsto %\...
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Why is this a good way to think geometrically about linear functionals?

In my textbook (Szekeres's A Course in Modern Mathematical Physics) the author writes Perhaps the best way to visualize a linear functional is as a set of parallel planes of vectors determined by $\...
EE18's user avatar
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What is the dual of integration over the unit interval?

Let $C[0,1]$ be the collection of continuous functions on the unit closed interval $[0,1]$. It is a vector space with respect to the usual addition and scaler multiplication. Let $\Phi:C[0,1]\to\...
govindah's user avatar
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If all compositions of a vector-valued function with its linear functionals are analytic, is the function itself analytic?

This is taken from Conway's A course in functional Analysis (p. 198, Exercise 4): Let $\mathscr{X}$ be a Banach space and $G \subset \mathbb{C}$ open. If $f: G \rightarrow \mathscr{X}$ is such that ...
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Quotient spaces and annihilators of inifinite dimensional vector spaces

Let $V$ be a vector space and $W$ be the dual space of $V$. For a subspace $U$ of $V$, we define $$ U^0 = \{f\in W \mid f(u)=0 \text{ for all } u\in U \}. $$ We are looking for an example in which $$ ...
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Dual of the annihilator isomorphic to quotient space

There are many resources online detailing the isomorphism between the annihilator $U^0:=\{f\in V':f(U)=0\}$ and the dual of the quotient space, $(V/U)':=Hom(V/U,\mathbb{R})$. However, my lecture notes ...
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How to understand the dual of determinants

We consider the determinant of a $n\times n$ square matrix as a linear transformation. $$\det : M_{n\times n}\to \mathbb{C}.$$ How to discribe its dual $\det ^*$?
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Weak continuity for a possibly nonlinear functional on a normed space

I am reading the fifth chapter ("Dual Spaces") from David Luenberger's Optimization by Vector Space Methods (1969). In Section 5.10, weak continuity for possibly nonlinear functionals on ...
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Uniqueness of Fourier transform of a measure [duplicate]

Let $C_0(\mathbb{R}^d)$ be the space of real-valued continuous functions on $\mathbb{R}^d$ that vanish at infinity. By Riesz–Markov–Kakutani representation theorem, the dual of $C_0(\mathbb{R}^d)$ is $...
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Definition of weak* continuity for possibly nonlinear functionals on normed dual spaces

I am reading the fifth chapter (on Dual Spaces) from David Luenberger's Optimization by Vector Space Methods. In Section 5.10, the author has defined weak continuity for possibly nonlinear functionals ...
AMathStudent's user avatar
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Let $f:V \times W \to \mathbb F$ and $s:V \to W^*, r:W \to V^*$. Find $[s]^B_{C^*}, [r]^C_{B^*}$.

Let $f:V \times W \to \mathbb F$ bilinear map and $s:V \to W^*, r:W \to V^*$ linear transformations s.t: $s(v)(w) = f(v,w)$ and $r(w)(v) = f(v,w), \forall v \in V$ and $\forall w \in W$. Let $B$ an ...
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What is dual space, in simple words?

I noticed that the rank of a tensor is a kind of "two-dimensional" property - the covariant components come first, then the contravariant. If I understand correctly, these components refer ...
Join the party P.A.R.T.Y.'s user avatar
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Confusion on proof of existence of ismophism between bilinear form on $L$ and dual space $\mathcal{L}(L,L^*)$

Theorem: There exists an isomorphism between the space of bilinear forms $\varphi$ on vector space $L$ and space $\mathcal{L}(L,L^*)$ of linear transformations $\mathcal{A}:L\mapsto L^*$. I'm ...
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Prove that the bilinear function $(l,x)$ gives a natural identification of $X$ with $X’’$.

I’m attempting a proof of Theorem 3 below. I would like to know if my proof is valid. If it is, is there a “cleaner” proof? If it is not valid, can you push me in the right direction? Please first ...
Paul Ash's user avatar
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Dual space of Sobolev spaces induced by Elliptic equations

Given a domain $\Omega \subset \mathbb{R}^n$, it is well-known that the dual space of the Sobolev space $W_0^{1,2}(\Omega)$ (also denoted by $H^1_0(\Omega)$ by many people) is, by definition, the ...
XIII's user avatar
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Relationship between the dual of a subspace and its annihilator

Let $V$ be a finite-dimensional vector space and let $W$ be a subspace of $V$. I shall use the following notation: $V^*$ will be the dual space of $V$, $Wº$ will be the annihilator of $W$, that is the ...
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Proving that a map is weak* continuous

Let $X$ be a compact space, $C(X)$ the space of continuous functions on $X$, and $\mathscr{M} = C(X)^*$, the dual of $C(X)$ which we identify with the space of all complex Baire measures. Let $$\...
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Proof: If T (T') is surjective then T' (T) is injective.

I am a little unsure about my proof. I wanted to ask if this proof is correct.I would appreciate any suggestions and improvements. Futhermore I dont yet fully understand why in (a) from the ...
Stippinator's user avatar
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Is there any bounded linear map $T:X\to X$ such that $T$ is injective, $T^{\ast}$ is not surjective and $R(T^{\ast})$ is closed?

Here $X$ is Banach space and $T^{\ast}$ is the continuous dual(adjoint) of $T,R(T^{\ast})$ stands for the range of $T^{\ast}.$ The following is what I get: If such a $T$ exists, then $X$ cannot be ...
Tiffany's user avatar
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Are the weakly bounded subsets of the double dual bounded for the natural topology?

I'm trying to get a better understanding of the topologies on the double dual of a Hausdorff locally convex space (l.s.c.). The following question has then come up, and I'm unable to find an answer. ...
user920957's user avatar
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Geometric Hahn-Banach theorem and linear operators [duplicate]

I've got this problem. $(X, ||\cdot||)$ is a normed vectorial space. Then, $l_0, l_1, ..., l_n$ are linear operators in $X^*$ (dual space) and for each $i\in \{0,1,...,n\}$, let $ker\:l_i = \{x\in X | ...
Tomas Rojas's user avatar
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Proving that a function $G: X\rightarrow \mathbb{R}^{n+1}$ is continuous.

$(X, ||\cdot||)$ is a normed vector space, and $l_0, l_1, ..., l_n$ are linear operators in $X^*$. And $G$ is defined like this. $G:X\rightarrow R^{n+1}: x\rightarrow (l_0(x), l_1(x), ..., l_n(x))$. ...
Tomas Rojas's user avatar
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How do I prove that a function is well defined? in my case, a continuous linear function.

This is my problem: Let $X$ and $Y$ be two normed vectorial spaces and let $L: X \rightarrow Y$ be a linear continuous functional. Let's now define $L^*: Y^* \rightarrow X^*$ like the the function ...
Tomas Rojas's user avatar
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Dual norm of integral vector norm

For $x\in\mathbb{C}^n$, consider the following norm: $\|x\|_{B} := \int_{B}|\langle x,y\rangle| \ d\mu(y),$ where $B:=\{y\in\mathbb{C}^n \ | \ \|y\|_2 \leq 1\}$ is the euclidean-norm ($\|\circ\|_2$) ...
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Necessary and sufficient condition to write a quadratic form on a finite-dimensional real vector space as a product of two linear functionals

I have come across a tricky linear algebra problem. We want to prove that a quadratic form $q$ on a finite dimensional real vector space $V$ can be expressed as $q(v) = f_1(v)f_2(v) \iff r + |\sigma| \...
Featherball's user avatar
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Show that multiplicative functional are extremal points in a $C^*$-algebra.

I found the following exercise in my lecture notes: Let $\mathcal{A}$ be a $C^*$-algebra. Then every multiplicative functional is an extremal point of $K_1^{\mathcal{A}'}(0)$. Other than writing $m = ...
julian2000P's user avatar
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Weak*-separability of the unit ball of $X’$ and density characters and cardinalities of $X$ and $X’$

Let $X$ be a Banach space, $X’$ be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about the density characters (which I’ll denote by $d$) and ...
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How can a vector dual space be used to model division?

A couple of months ago I posted a question about how to formally define systems of units, and someone posted this fascinating response, explaining that each base unit can be the basis of a single ...
Mikayla Eckel Cifrese's user avatar
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If $L_n \rightarrow L \in X^*$ in the weak* sense and $x_n \rightarrow x$ in norm, does $L_n(x_n) \rightarrow L(x)$?

Let $X$ be a Banach space and $X^*$ its dual. Let $\{L_n\}$ be a sequence in $X^*$ such that $L_n \rightarrow L$ in the weak* sense, and suppose $\{x_n\}$ is a sequence in $X$ such that $x_n \...
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Containment of kernels of continuous seminorms

Let $X$ be a locally convex space, whose topology is defined by a family of seminorms $\mathcal{P}$. Fact. For every continuous seminorm $q: X\to \mathbb{F}$ and positive number $\epsilon>0$, the ...
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